Flavor dependence of the S-parameter

We extend the results of Sannino (2010) [1] by computing the S-parameter at two loops in the perturbative region of the conformal window. Consistently using the expression for the location of the infrared fixed point at the two-loop order we express the S-parameter in terms of the number of flavors, colors and matter representation. We show that S, normalized to the number of flavors, increases as we decrease the number of flavors. Our findings support the conjecture presented in Sannino (2010) [1] according to which the normalized value of the S-parameter at the upper end of the conformal window constitutes the lower bound across the entire phase diagram for the given underlying asymptotically free gauge theory. We also show that the non-trivial dependence on the number of flavors merges naturally with the non-perturbative estimate of the S-parameter close to the lower end of the conformal window obtained using gauge duality (Sannino, 2010) [2]. Our results are natural benchmarks for lattice computations of the S-parameter for vector-like gauge theories.